Related papers: Several consequences of adding or removing an edge…

A constructive proof is given to the fact that any ergodic Markov chain can be realized as a random walk subject to a synchronizing road coloring. Redundancy (ratio of extra entropy) in such a realization is also studied.

We develop efficient and effective strategies for the update of Katz centralities after node and edge removal in simple graphs. We provide explicit formulas for the ``loss of walks" a network suffers when nodes/edges are removed, and use…

We suggest a model for data losses in a single node of a packet-switched network (like the Internet) which reduces to one-dimensional discrete random walks with unusual boundary conditions. The model shows critical behavior with an abrupt…

Given a connected graph $G$, Kemeny's constant $\mathcal{K}({G})$ measures the average travel time for a random walk to reach a randomly selected vertex. It is known that when an edge is added to $G$, the value of Kemeny's constant may…

A model for growing networks is introduced, having as a main ingredient that new nodes are attached to the network through one existing node and then explore the network through the links of the visited nodes. From exact calculations of two…

There has been a considerable amount of interest in recent years on the robustness of networks to failures. Many previous studies have concentrated on the effects of node and edge removals on the connectivity structure of a static network;…

Graphs are pervasive in our everyday lives, with relevance to biology, the internet, and infrastructure, as well as numerous other applications. It is thus necessary to have an understanding as to how quickly a graph disintegrates, whether…

Most empirical studies of networks assume that the network data we are given represent a complete and accurate picture of the nodes and edges in the system of interest, but in real-world situations this is rarely the case. More often the…

The impact of finite cycles on the phantom modulus in an otherwise perfect network is computed exactly. It is shown that pending cycles reduce the phantom modulus of the network by $kT/V$ independent of junction functionality. The…

Network games are widely used as a model for selfish resource-allocation problems. In the classical model, each player selects a path connecting her source and target vertices. The cost of traversing an edge depends on the {\em load};…

A short proof of the equivalence of the recurrence of non-backtracking random walk and that of simple random walk on regular infinite graphs is given. It is then shown how this proof can be extended in certain cases where the graph in…

Spreading processes on graphs are a natural model for a wide variety of real-world phenomena, including information spread over social networks and biological diseases spreading over contact networks. Often, the networks over which these…

We study directed random graphs (random graphs whose edges are directed) as they evolve in discrete time by the addition of nodes and edges. For two distinct evolution strategies, one that forces the graph to a condition of near acyclicity…

Connectivity is a fundamental structural feature of a network that determines the outcome of any dynamics that happens on top of it. However, an analytical approach to obtain connection probabilities between nodes associated to paths of…

A second-order random walk on a graph or network is a random walk where transition probabilities depend not only on the present node but also on the previous one. A notable example is the non-backtracking random walk, where the walker is…

The exact mean time between encounters of a given particle in a system consisting of many particles undergoing random walks in discrete time is calculated, on both regular and complex networks. Analytical results are obtained both for…

In this paper we introduce the notion of Random Walk in Changing Environment - a random walk in which each step is performed in a different graph on the same set of vertices, or more generally, a weighted random walk on the same vertex and…

We present the analytical and numerical results of a random walk on the family of small-world graphs. The average access time shows a crossover from the regular to random behavior with increasing distance from the starting point of the…

Random walks find applications in many areas of science and are the heart of essential network analytic tools. When defined on temporal networks, even basic random walk models may exhibit a rich spectrum of behaviours, due to the…

In spatial networks vertices are arranged in some space and edges may cross. When arranging vertices in a 1-dimensional lattice edges may cross when drawn above the vertex sequence as it happens in linguistic and biological networks. Here…